Keywords: Timoshenko's flexural waves, improved third-order mass matrix, velocity dispersion, numerical simulation.

The objective of this paper is presenting an improved third-order mass matrix to solve Timoshenko's flexural wave equation. The dispersive velocity of the propagation properties of the finite element solution by using semi-discretizations of Timoshenko's flexural wave equation, as well as bifurcations of the first- and second-spectra flexural waves illustrate a very good performance of the proposed consistent mass model included for a low frequency regime. Classic two-node cubic elements are taken into account. The Newmark average acceleration step-by-step integration method is used for integration in time domain. The attention is also devoted to the evanescent first- and second-spectra waves.

Since the classical Bernoulli-Euler theory of flexural wave propagation has been recognized as inadequate to solve wave propagation of higher modes [1], various efforts have been devoted to its improvement based on the Timoshenko beam theory.

During the recent decades, a number of mass matrices to solve Timoshenko's flexural wave propagation model have been proposed in the literature [2]. This paper presents an improved third-order mass matrix to solve Timoshenko's flexural wave propagation problems in which dynamic equilibrium is considered [3]. The best attributes of this third-order mass matrix may be its low numeric velocity dispersion in comparison with classical consistent mass matrix even for low frequency wave propagation.

In order to obtain third-order convergence the present formulation considers initially the Taylor expansion for displacement and bending rotation (the element length is the space variable increment). The new consistent mass matrix is then obtained by annulling the terms of the Taylor expansion lower than third-order. On the other hand, the velocity dispersion analysis is studied by examining the discrete equilibrium equation of motion (equilibrium of a generic node) worked in terms of the numerical wave motion solution.

The wave propagation on a typical wide flange shape beam is considered as a numerical application. By examining the main results (numerical wave number and eigen-vector components) one can observe that the proposed consistent mass matrix presents velocity dispersion (global error) similar to the classical one.

The new proposed mass matrix can be a good mathematical tool to study Timoshenko's flexural wave propagation problems as it presents elements expressed by single terms.

1

J.E. Laier, "Mass lumping, dispersive properties and bifurcation of Timoshenko's flexural waves", Adv Eng Software, 33, 605-610, 2007. doi:10.1016/j.advengsoft.2006.08.018

2

H. Ahmadian, M.I. Friswell, J.E. Mottershead, "Minimization of the Discretization Error in Mass and Stiffness formulations by an Inverse Method", Int. J. Numer. Meth. Engng., 45, 371-387, 1998. doi:10.1002/(SICI)1097-0207(19980130)41:2<371::AID-NME288>3.3.CO;2-I

3

Z.P. Bazant, "Spurious reflection of elastic waves in nonuniform finite element grids", Comp Meth Appl Mech Eng, 16, 91-100, 1978. doi:10.1016/0045-7825(78)90035-X

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